Mappings and flatness in non-smooth geometry

Classical calculus concerns itself with smooth functions, curves, and surfaces inside our standard Euclidean geometry. Nowadays, many problems in pure and applied mathematics benefit from extending calculus tools beyond these smooth, Euclidean contexts. This includes the study of non-smooth functions and sets, abstract geometries different from our own, and fractal spaces with interesting behavior at many scales. The project aims to understand these objects in a number of ways: by studying the ways curves can pass through them, by decomposing them into simpler pieces, by embedding or folding them into classical geometries, or by approximating them with linear objects – measuring how “flat” they are at various locations and scales. Non-smooth analysis and geometry are important in many areas of pure and applied mathematics and computer science, since non-smooth problems arise in studying large data sets, in computational questions, and as limiting cases of smooth problems. The project also emphasizes techniques and results that are "quantitative": providing guaranteed estimates independent of the particular function or geometry being studied. The project's broader impacts include organizational work in the analysis community, focusing on early-career researchers, and impact on the research environment in the PI's home department through invited speakers and improved advising and mentoring. In general, the project contributes to a more robust understanding of complex and high-dimensional mathematical objects that appear in many problems. More specifically, the PI is continuing his study of newly constructed “coarse” and “pointwise” tangent fields for non-smooth sets in high-dimensional ambient spaces, analogous in some ways to the tangent planes of a smooth surface. These provide new ways to describe the structure of non-smooth sets. Properties and applications of this construction are related to questions, also under investigation in the project, about when one can project, embed, or fold high-dimensional sets into a lower-dimensional space without too much distortion of the geometry. Flatness properties of sets and mappings also arise in connection with parametrization problems for manifolds (when is a topological sphere bi-Lipschitz or quasisymmetric to a standard sphere?) and decomposition problems for mappings (when can we break a complex geometric mapping into a sequence of steps with small distortion, or break its domain into sets on which it acts more simply?). Related approximation, parametrization, and embedding questions also apply even for abstract metric spaces, which are also within the scope of the project. The PI studies these questions with a mixture of techniques from geometric measure theory, analysis on metric spaces, quasiconformal geometry, and geometric topology. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2553261 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Guy David | Institution: Ball State University, MUNCIE, IN | Award Amount: $125,349 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2553261 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2553261.html